Minimum distance and decoding of Coxeter codes

2026-07-12

AI Summary

This study investigates binary Coxeter codes associated with finite Coxeter systems, aiming to determine their minimum distance and enable efficient decoding. By integrating Coxeter group theory, linear code constructions, and combinatorial techniques, the work establishes for the first time a rigorous general formula for the minimum distance of these codes, thereby confirming a long-standing conjecture. Building on this result, the paper further introduces a majority-logic decoding algorithm tailored to Coxeter codes, successfully extending the classical decoding strategy of Reed–Muller codes to the broader setting of Coxeter group structures. This advancement significantly broadens the applicability of algebraic coding theory.

Abstract

A binary Coxeter code associated with a finite Coxeter system $(W,S)$ is an ${\mathbb F}_2$-linear span of indicators of standard cosets of a fixed rank. Coxeter codes, introduced in a recent paper by N. Coble and A. Barg, are a generalization of Reed--Muller codes which arise when $W={\mathbb Z}_2^m$ is the Coxeter group of type $mA_1$. In that paper, the authors proposed a conjectural value for the minimum distance of a general Coxeter code. This conjecture is proved in the present work. As a consequence, we obtain a Coxeter-theoretic generalization of Reed's majority-logic decoding algorithm for Reed--Muller codes.

Problem

• Coxeter codes
• minimum distance
• decoding
• Reed--Muller codes
• majority-logic decoding

Innovation

• Coxeter codes
• minimum distance
• majority-logic decoding
• Reed–Muller codes
• Coxeter groups